Final answer to the problem
Step-by-step Solution
How should I solve this problem?
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- Solve using L'Hôpital's rule
- Solve without using l'Hôpital
- Solve using limit properties
- Solve using direct substitution
- Solve the limit using factorization
- Solve the limit using rationalization
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
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Plug in the value $0$ into the limit
The cosine of $0$ equals $1$
Multiply $-1$ times $1$
Subtract the values $1$ and $-1$
Calculate the power $0^2$
If we directly evaluate the limit $\lim_{x\to0}\left(\frac{1-\cos\left(x\right)}{x^2}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form
We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
Find the derivative of the numerator
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$
Any expression multiplied by $1$ is equal to itself
Find the derivative of the denominator
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
After deriving both the numerator and denominator, and simplifying, the limit results in
Plug in the value $0$ into the limit
The sine of $0$ equals $0$
Multiply $2$ times $0$
If we directly evaluate the limit $\lim_{x\to0}\left(\frac{\sin\left(x\right)}{2x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form
We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
Find the derivative of the numerator
The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$
Find the derivative of the denominator
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
After deriving both the numerator and denominator, and simplifying, the limit results in
Evaluate the limit $\lim_{x\to0}\left(\frac{\cos\left(x\right)}{2}\right)$ by replacing all occurrences of $x$ by $0$
The cosine of $0$ equals $1$
